FRG: COLLABORATIVE RESEARCH: Super Approximation and Thin Groups, with Applications to Geometry, Groups, and Number Theory

FRG：协作研究：超逼近和薄群，及其在几何、群和数论中的应用

• 批准号: 1463940
• 负责人: Alex Kontorovich
• 金额: $ 34.12万
• 依托单位: Rutgers University New Brunswick
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-07-01 至 2019-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1463940&HistoricalAwards=false
• 关键词: FRG; COLLABORATIVE; RESEARCH; Super; Approximation

One of the most fundamental objects in mathematics is the "group," a set with a rule analogous to multiplication for combining elements of the set. Groups can be viewed as precise ways to capture the symmetries of sets, shapes, and other mathematical objects. The last decade has seen an explosion of activity that can be viewed through the lens of what is called Super Approximation. Very roughly speaking, this refers to the idea that walking around randomly on the points of a group mixes things up very rapidly. The growth of this subject was swiftly followed by a variety of applications to geometry, groups, and number theory. The striking symbiosis of the resultant collection of problems and fields has inspired this team of researchers to unify and more deeply connect these and related themes of research, in order to make further advances. The Principal Investigators, as well as their postdocs and students, will work on a variety of projects concerned with the group theoretic, geometric, and number theoretic aspects of Super Approximation. Exponential sums and "Affine Sieve" methods will be developed further to "Local-Global" settings, including attacks on McMullen's Arithmetic Chaos Conjecture and Zaremba's Conjecture. The PIs will furthermore explore the use of privileged circle and sphere packings to understand the construction of trace (and invariant trace) fields for hyperbolic manifolds, a long-standing and almost completely untouched aspect of the theory. Moreover, the PIs will investigate the construction of interesting subgroups of arithmetic lattices via geometric deformations, as well as study problems in combinatorics and computer science.

数学中最基本的对象之一是“群”，这是一个具有类似于乘法的规则的集合，用于组合集合的元素。 群可以被视为捕捉集合、形状和其他数学对象的对称性的精确方法。过去十年中，活动呈爆炸式增长，可以通过所谓的“超级近似”的视角来观察。粗略地说，这是指在一组点上随机走动会很快地将事情混合在一起的想法。该学科的发展很快就出现了几何、群和数论的各种应用。由此产生的问题和领域的惊人共生激发了这组研究人员将这些和相关的研究主题统一并更深入地联系起来，以取得进一步的进展。首席研究员以及他们的博士后和学生将致力于各种与超逼近的群论、几何和数论方面相关的项目。指数和和“仿射筛”方法将进一步发展到“局部-全局”设置，包括对麦克马伦算术混沌猜想和扎伦巴猜想的攻击。 PI 将进一步探索使用特权圆和球填充来理解双曲流形的迹（和不变迹）场的构造，这是该理论的一个长期存在且几乎完全未触及的方面。此外，PI 将研究通过几何变形构建有趣的算术格子组，以及研究组合学和计算机科学中的问题。